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IMPORTANT CHANGE TO CONTEST RULES FOR MCM/ICM 2024:
Teams (Student or Advisor) are now required to submit an electronic copy (summary sheet and solution) of their solution paper by email to . Your email MUST be received at COMAP by the submission deadline of 8:00 PM EST, February 13, 2024. Teams are free to choose between MCM Problem A, MCM Problem B or ICM Problem C.
COMAP Mirror Site: For more in:
MCM: The Mathematical Contest in Modeling
ICM: The Interdisciplinary Contest in Modeling
2024 Contest Problems
MCM PROBLEMS
PROBLEM A: The Leaves of a Tree
"How much do the leaves on a tree weigh?" How might one estimate the actual weight of the leaves (or for that matter any other parts of the tree)? How might one classify leaves? Build a mathematical model to describe and classify leaves. Consider and answer the following:
• Why do leaves have the various shapes that they have?
• Do the shapes “minimize” overlapping individual shadows that are cast, so as to maximize exposure? Does the distribution of leaves within the “volume” of the tree and its branches effect the shape?
• Speaking of profiles, is leaf shape (general characteristics) related to tree pro structure?
• How would you estimate the leaf mass of a tree? Is there a correlation between the leaf mass and the size characteristics of the tree (height, mass, volume defined by the profile)?
In addition to your one page summary sheet prepare a one page letter to an editor of a scientific journal outlining your key findings.
2024美赛A题：一棵树的叶子
（数学中国翻译）
“一棵树的叶子有多重？”怎么能估计树的叶子（或者树的任何其它部分）的实际重量？怎样对叶子进行分类？建立一个数学模型来对叶子进行描述和分类。模型要考虑和回答下面的问题：
• 为什么叶子具有各种形状？
• 叶子之间是要将相互重叠的部分最小化，以便可以最大限度的接触到阳光吗？树叶的分布以及树干和枝杈的体积影响叶子的形状吗？
就轮廓来讲，叶形（一般特征）是和树的轮廓以及分枝结构有关吗？
• 你将如何估计一棵树的叶子质量？叶子的质量和树的尺寸特征（包括和外形轮廓有关的高度、质量、体积）有联系吗？
除了你的一页摘要以外，给科学杂志的编辑写一封信，阐述你的主要发现。
PROBLEM B: Camping along the Big Long River
Visitors to the Big Long River (225 miles) can enjoy scenic views and exciting white water rapids. The river is inaccessible to hikers, so the only way to enjoy it is to take a river trip that requires several days of camping. River trips all start at First Launch and exit the river at Final Exit, 225 miles downstream. Passengers take either oar- powered rubber rafts, which travel on average 4 mph or motorized boats, which travel on average 8 mph. The trips range from 6 to 18 nights of camping on the river, start to finish.. The government agency responsible for managing this river wants every trip to enjoy a wilderness experience, with minimal contact with other groups of boats on the river. Currently, X trips travel down the Big Long River each year during a six month period (the rest of the year it is too cold for river trips). There are Y camp sites on the Big Long River, distributed fairly uniformly throughout the river corridor. Given the rise in popularity of river rafting, the park managers have been asked to allow more trips to travel down the river. They want to determine how they might schedule an optimal mix of trips, of varying duration (measured in nights on the river) and propulsion (motor or oar) that will utilize the campsites in the best way possible. In other words, how many more boat trips could be added to the Big Long River’s rafting season? The river managers have hired you to advise them on ways in which to develop the best schedule and on ways in which to determine the carrying capacity of the river, remembering that no two sets of campers can occupy the same site at the same time. In addition to your one page summary sheet, prepare a one page memo to the managers of the river describing your key findings.
问题B：沿长江大露营
到Big Long River（225英里）游玩的游客可以享受那里的风景和振奋人心的急流。远足者没法到达这条河，唯一去的办法是漂流过去。这需要几天的露营。河流旅行始于First Launch，在 Final Exit结束，共225英里的顺流。旅客可以选择依靠船桨来前进的橡皮筏，它的速度是4英里每小时，或者选择8英里每小时的摩托船。旅行从开始到结束包括大约6到18个晚上的河中的露营。负责管理这条河的政府部门希望让每次旅行都能尽情享受野外经历，同时能尽量少的与河中其他的船只相遇。当前，每年经过Big Long河的游客有X组，这些漂流都在一个以6个月长短的时期内进行，一年中的其他月份非常冷，不会有漂流。在Big Long上有Y处露营地点，平均分布于河廊。随着漂流人数的增加，管理者被要求应该允许让更多的船只漂流。他们要决定如何来安排最优的方案：包括旅行时间（以在河上的夜晚数计算）、选择哪种船（摩托还是桨船），从而能够最好地利用河中的露营地。换句话说，Big Long River在漂流季节还能增加多少漂流旅行数？管理者希望你能给他们最好的建议，告诉他们如何决定河流的容纳量，记住任两组旅行队都不能同时占据河中的露营地。此外，在你的摘要表一页，准备一页给管理者的备忘录，用来描述你的关键发现。
ICM PROBLEM
PROBLEM C: Modeling for Crime Busting
Click the title below to download a ZIP the 2024 ICM Problem.
Your ICM submission should consist of a 1 page Summary Sheet and your solution cannot exceed 20 pages for a maximum of 21 pages.
2024 ICM Problem
Modeling for Crime Busting
Your organization, the Intergalactic Crime Modelers (ICM), is investigating a conspiracy to commit a
criminal act. The investigators are highly confident they know several members of the conspiracy, but
hope to identify the other members and the leaders before they make arrests. The conspirators and
the possible suspected conspirators all work for the same company in a large office complex. The
company has been growing fast and making a name for itself in developing and marketing computer
software for banks and credit card companies. ICM has recently found a small set of messages from a
group of 82 workers in the company that they believe will help them find the most likely candidates for
the unidentified co‐conspirators and unknown leaders. Since the message traffic is for all the office
workers in the company, it is very likely that some (maybe many) of the identified communicators in the
message traffic are not involved in the conspiracy. In fact, they are certain that they know some people
who are not in the conspiracy. The goal of the modeling effort will be to identify people in the office
complex who are the most likely conspirators. A priority list would be ideal so ICM could investigate,
place under surveillance, and/or interrogate the most likely candidates. A discriminate line separating
conspirators from non‐conspirators would also be helpful to distinctly categorize the people in each
group. It would also be helpful to the DA’s office if the model nominated the conspiracy leaders.
Before the data of the current case are given to your crime modeling team, your supervisor gives you
the following scenario (called Investigation EZ) that she worked on a few years ago in another city. Even
though she is very proud of her work on the EZ case, she says it is just a very small, simple example, but
it may help you understand your task. Her data follow:
The ten people she was considering as conspirators were: Anne#, Bob, Carol, Dave*, Ellen, Fred,
George*, Harry, Inez, and Jaye#. (* indicates prior known conspirators, # indicate prior known nonconspirators)
Chronology of the 28 messages that she had for her case with a code number for the topic of each
message that she assigned based on her analysis of the message:
Anne to Bob: Why were you late today? (1)
Bob to Carol: That darn Anne always watches me. I wasn't late. (1)
Carol to Dave: Anne and Bob are fighting again over Bob's tardiness. (1)
Dave to Ellen: I need to see you this morning. When can you come by? Bring the budget files. (2)
Dave to Fred: I can come by and see you anytime today. Let me know when it is a good time. Should I
bring the budget files? (2)
Dave to George: I will see you later ‐‐‐ lots to talk about. I hope the others are ready. It is important to
get this right. (3)
Harry to George: You seem stressed. What is going on? Our budget will be fine. (2) (4)
Inez to George: I am real tired today. How are you doing? ( 5)
Jaye to Inez: Not much going on today. Wanna go to lunch today? (5)
Inez to Jaye: Good thing it is quiet. I am exhausted. Can't do lunch today ‐‐‐ sorry! (5)
George to Dave: Time to talk ‐‐‐ now! (3)
Jaye to Anne: Can you go to lunch today? (5)
Dave to George: I can't. On my way to see Fred. (3)
George to Dave: Get here after that. (3)
Anne to Carol: Who is supposed to watch Bob? He is goofing off all the time. (1)
Carol to Anne: Leave him alone. He is working well with George and Dave. (1)
George to Dave: This is important. Darn Fred. How about Ellen? (3)
Ellen to George: Have you talked with Dave? (3)
George to Ellen: Not yet. Did you? (3)
Bob to Anne: I wasn't late. And just so you know ‐‐‐ I am working through lunch. (1)
Bob to Dave: Tell them I wasn't late. You know me. (1)
Ellen to Carol: Get with Anne and figure out the budget meeting schedule for next week and help me
calm George. (2)
Harry to Dave: Did you notice that George is stressed out again today? (4)
Dave to George: Darn Harry thinks you are stressed. Don't get him worried or he will be nosing around.
(4)
George to Harry: Just working late and having problems at home. I will be fine. (4)
Ellen to Harry: Would it be OK, if I miss the meeting today? Fred will be there and he knows the
budget better than I do. (2)
Harry to Fred: I think next year's budget is stressing out a few people. Maybe we should take time to
reassure people today. (2) (4)
Fred to Harry: I think our budget is pretty healthy. I don't see anything to stress over. (2)
END of MESSAGE TRAFFIC
Your supervisor points outs that she assigned and coded only 5 different topics of messages: 1) Bob's
tardiness, 2) the budget, 3) important unknown issue but assumed to be part of conspiracy, 4) George's
stress, and 5) lunch and other social issues. As seen in the message coding, some messages had two
topics assigned because of the content of the messages.
The way your supervisor analyzed her situation was with a network that showed the communication
links and the types of messages. The following figure is a model of the message network that resulted
with the code for the types of messages annotated on the network graph.
Figure 1: Network of messages from EZ Case
Your supervisor points out that in addition to known conspirators George and Dave, Ellen and Carol
were indicted for the conspiracy based on your supervisor's analysis and later Bob self‐admitted his
involvement in a plea bargain for a reduced sentence, but the charges against Carol were later dropped.
Your supervisor is still pretty sure Inez was involved, but the case against her was never established.
Your supervisor's advice to your team is identify the guilty parties so that people like Inez don't get off,
people like Carol are not falsely accused, and ICM gets the credit so people like Bob do not have the
opportunity to get reduced sentences.
The current case:
Your supervisor has put together a network‐like database for the current case, which has the same
structure, but is a bit larger in scope. The investigators have some indications that a conspiracy is taking
place to embezzle funds from the company and use internet fraud to steal funds from credit cards of
people who do business with the company. The small example she showed you for case EZ had only 10
people (nodes), 27 links (messages), 5 topics, 1 suspicious/conspiracy topic, 2 known conspirators, and 2
known non‐conspirators. So far, the new case has 83 nodes, 400 links (some involving more than 1
topic), over 21,000 words of message traffic, 15 topics (3 have been deemed to be suspicious), 7 known
conspirators, and 8 known non‐conspirators. These data are given in the attached spreadsheet files:
Names.xls, Topics.xls, and Messages.xls. Names.xls contains the key of node number to the office
workers' names. Topics.xls contains the code for the 15 topic numbers to a short description of the
topics. Because of security and privacy issues, your team will not have direct transcripts of all the
message traffic. Messages.xls provides the links of the nodes that transmitted messages and the topic
code numbers that the messages contained. Several messages contained up to three topics. To help
visualize the message traffic, a network model of the people and message links is provided in Figure 2.
In this case, the topics of the messages are not shown in the figure as they were in Figure 1. These topic
numbers are given in the and described in Topics.xls.
Figure 2: Visual of the network model of the 83 people (nodes) and 400 messages between these
people (links).
Requirements:
Requirement 1: So far, it is known that Jean, Alex, Elsie, Paul, Ulf, Yao, and Harvey are conspirators.
Also, it is known that Darlene, Tran, Jia, Ellin, Gard, Chris, Paige, and Este are not conspirators. The
three known suspicious message topics are 7, 11, and 13. There is more detail about the topics in file
Topics.xls. Build a model and algorithm to prioritize the 83 nodes by likelihood of being part of the
conspiracy and explain your model and metrics. Jerome, Delores, and Gretchen are the senior
managers of the company. It would be very helpful to know if any of them are involved in the
conspiracy.
Requirement 2: How would the priority list change if new information comes to light that Topic 1 is
also connected to the conspiracy and that Chris is one of the conspirators?
Requirement 3: A powerful technique to obtain and understand text information similar to this
message traffic is called semantic network analysis; as a methodology in artificial intelligence and
computational linguistics, it provides a structure and process for reasoning about knowledge or
language. Another computational linguistics capability in natural language processing is text analysis.
For our crime busting scenario, explain how semantic and text analyses of the content and context of
the message traffic (if you could obtain the original messages) could empower your team to develop
even better models and categorizations of the office personnel. Did you use any of these capabilities on
the topic descriptions in to enhance your model?
Requirement 4: Your complete report will eventually go to the DA so it must be detailed and clearly
state your assumptions and methodology, but cannot exceed 20 pages of write up. You may include
your programs as appendices in separate files that do not count in your page restriction, but including
these programs is not necessary. Your supervisor wants ICM to be the world's best in solving whitecollar,
high‐tech conspiracy crimes and hopes your methodology will contribute to solving important
cases around the world, especially those with very large databases of message traffic (thousands of
people with tens of thousands of messages and possibly millions of words). She specifically asked you
to include a discussion on how more thorough network, semantic, and text analyses of the message
contents could help with your model and recommendations. As part of your report to her, explain the
network modeling techniques you have used and why and how they can be used to identify, prioritize,
and categorize similar nodes in a network database of any type, not just crime conspiracies and message
data. For instance, could your method find the infected or diseased cells in a biological network where
you had various kinds of image or chemical data for the node
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